75.18.10 problem 599

Internal problem ID [17027]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Initial value problem. Exercises page 140
Problem number : 599
Date solved : Monday, March 31, 2025 at 03:38:32 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=4 x \cos \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 13

ode:=diff(diff(y(x),x),x)+y(x) = 4*x*cos(x); 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = x \left (x \sin \left (x \right )+\cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 86

ode=D[y[x],{x,2}]+y[x]==4*x*Cos[x]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \sin (x) \left (-\int _1^04 \cos ^2(K[2]) K[2]dK[2]\right )+\sin (x) \int _1^x4 \cos ^2(K[2]) K[2]dK[2]-\cos (x) \int _1^0-2 K[1] \sin (2 K[1])dK[1]+\cos (x) \int _1^x-2 K[1] \sin (2 K[1])dK[1]+\sin (x) \]

✓ Sympy. Time used: 0.117 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*cos(x) + y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{2} \sin {\left (x \right )} + x \cos {\left (x \right )} \]